modeling of electric arcs: a study of the non-convective case with
TRANSCRIPT
J. Plasma Physics: page 1 of 15. c© Cambridge University Press 2013
doi:10.1017/S0022377813000317
1
Modeling of electric arcs: A study of the non-convectivecase with strong coupling
D. W R I G H T1, P. D E L M O N T2 and M. T O R R I L H O N 2
1Seminar for Applied Mathematics, ETH Zurich, Ramistr. 101, 8092 Zurich, Switzerland2MathCCES, RWTH Aachen University, Schinkelstr. 2, 52062 Aachen, Germany
(Received 17 January 2013; revised 15 February 2013; accepted 18 February 2013)
Abstract. In this paper, we investigate a mathematical model for electric arcs. Themodel is based on the equations of magnetohydrodynamics, where the flow andheat transfer in a plasma is coupled to electrodynamics. Our approach neglectsconvection and yields a reaction–diffusion model that includes only the corephenomena of electric arcs: Ohmic heating and nonlinear electric conductivity.The equations exhibit interesting mathematical properties like non-unique steadystates and instabilities that can be linked to electric arc properties. Additionally,a 3D axisymmetric simulation of the creation and extinction of an electric arc ispresented based on a strongly coupled numerical algorithm for the non-convectivemodel. The approach is especially suited for high-current arcs where strong couplingbecomes necessary.
1. IntroductionAn electric arc is a discharge phenomenon in whichcurrent is conducted through a hot-ionized gas betweenan anode and a cathode. Typically, the temperature ofthe cathode is high enough so that electrons are emittedwith almost no additional external force. The ionized gasforms a plasma which interacts with electric and mag-netic fields of the current. Electric arcs occur increasinglyin various industrial applications from melting, drilling,lighting to circuit breakers and electric thrusters. Hence,mathematical plasma modeling and plasma simulationsare becoming increasingly important (for a review ofthe matter, see Gleizes et al. 2005). Electric arcs havebeen studied from the fundamental and experimentalphysics perspectives in the middle of the 20th centurywith classical publications (e.g. Cassie 1939; Mayr 1943;Maecker 1951; Mayr 1955; Finkelnburger and Maecker1956; Rieder 1967). More modern, full arc simulationscan be found for example in Kosse et al. (2007) andZheng et al. (2004).
Some of the strongest electric arcs in technical pro-cesses are found in high performance circuit breakerswith currents up to 200 kA and energies up to 1500MW (see Van Der Sluis 2001; Garzon 2002; Zehnderet al. 2002). The largest of these breakers are installedin power plants to switch fault currents in emergenciesin order to prevent damage to the interior of the plantresulting from an overload. Since it is impossible toprevent the creation of the electric arc when separatingthe contacts, a circuit breaker is designed to extinguishthe arc as fast as possible once it is created. The twodissertations Huguenot (2008) and Kumar (2009) aim
at the detailed and full-scale simulation of high-currentelectric arcs in high performance circuit breakers. Inthe theses, some proof of concepts could be given,but it also became clear that the simulation of thismulti-physics process requires additional basic researchin mathematical modeling and numerical analysis.
This paper starts with the equations of single-fluid realplasma magnetohydrodynamics (MHD) to study thefundamental mathematical mechanisms of electric arcs.As first step, a non-convective model is derived from theMHD equations under the assumption that the motionof the plasma is negligible. A future paper will extend thepresent results to plasma convection. The non-convectivemodel is based on nonlinear electric conductivity andOhmic heating as the fundamental mechanisms in theelectric arc. The study of axisymmetric arcs then allowsus to study the stability of electric arc. In that way, thecreation and extinction of the arc can be viewed as aninterplay of stable and unstable solutions to a system ofreaction–diffusion equations, see also Torrilhon (2007).A 3D axisymmetric simulation demonstrates the useful-ness of the model in more realistic geometries. Specialattention is paid to the possibility to switch from voltage-driven to current-driven according to requirements fromthe external electric network. This requires to change theboundary conditions of the magnetic field accordingly.
Typically, in arc simulations, the equations of elec-trodynamics are coupled weakly to the flow and heattransfer, that is, the systems are often solved in asplit way and Lorentz force and Ohmic heating trans-ferred as external productions per time-step. This pa-per uses a strong coupling in which heat transfer and
2 D. Wright et al.
electrodynamics are solved in a fully coupled way usinga single implicit time integration. In this approach, thesplitting is not introduced between heat conducting flowand electrodynamics, but between Ohmic heating andcurrent diffusion on the one hand and non-dissipativeeffects like the flow on the other hand. This followsthe argument that the arc is driven by Ohmic heatingand nonlinear current diffusion, while the flow can beconsidered a reaction. As a result, the numerical methodbecomes more stable and robust especially for very highcurrent situations.
As mentioned above, the presented model needs tobe coupled to the plasma flow. However, the non-convective model can help to understand full-scale arcsimulations better and may have relevant applications.For instance, initialization with a burning arc provedmost difficult in Huguenot (2008) and Kumar (2009). Aphysical approach to create the electric arc as presentedin this paper may be beneficial in these simulations. Themodel presented in this paper can also be coupled toan external electric network (see Delmont and Torrilhon2012b) and used to advance black-box Cassie–Mayrmodels, as in Tseng et al. (1997) and Maximov et al.(2009). First convective simulations have also been pub-lished (see Delmont and Torrilhon 2012a).
The rest of the paper is organized as follows. Wesimplify the equations of dissipative and resistive MHDfor the case of a non-convective arc and discuss themodeling and scaling in the next section. Then, Sec.3 specializes the equations for an infinite axisymmetricarc column and we study steady (Sec. 4) and unsteady(Sec. 5) solutions and their stability. Section 6 describes3D axisymmetric simulations. The paper ends with aconclusion.
2. MagnetohydrodynamicsIn MHD, plasma is described on the basis of a singlegas that models the electrical, chemical and thermody-namical behavior of the mixture of ions and electronsas a whole (see Goedbloed and Poedts 2004). MHDconsiders the fields of density ρ, velocity v, temperatureT , and magnetic field B.
2.1. Equations
The equations of MHD are given by the conservationlaws of mass, momentum and energy for the plasma andMaxwell’s equation in eddy current approximation, thatis, neglecting the displacement current. They read
∂tρ+ div ρv = 0,
∂tρv + div (ρvv + p I +Π) = j × B,
∂tρ(ε+ 1
2v2
)+ div
(ρv
(ε+ 1
2v2 + p
)+Π · v + q
)= j · E + r,
∂tB + curlE = 0,
curlB = μ0j, (2.1)
where j is the electric current density in the plasma and Eis the electric field. The permeability constant of vacuumis μ0 = 1.26 × 10−6V s (A m)−1. While the last equation(Ampere’s law) can be viewed as an equation for thecurrent, we need a closure relation for the electric field.This can be obtained from Ohm’s law
j = σ(T ) (E + v × B) , (2.2)
which gives
E = B × v +1
σ(T )j = B × v +
1
σ(T )curlB (2.3)
to enter the induction equation for B in (2.1). Theelectric conductivity σ(T ) depends on temperature andis discussed later. The influence of the electric part onthe fluid equations of (2.1) comes from the Lorentz forceand Ohmic heating in the momentum equation and en-ergy equation, respectively. The fluid flow influences theequations for the electromagnetic field via the velocityin (2.3).
The fluid equations in (2.1) require additional closurerelations for stress tensor Π , heat flux q, and internalenergy ε, as well as the radiation r. We assume thesimplest thermodynamic laws
Π = −2μ(T ) grad v, q = −λ(T ) gradT , (2.4)
where grad v represents the traceless and symmetric partof the gradient. The viscosity coefficient μ(T ) and heatconductivity λ(T ) both depend on temperature. Theinternal energy follows from the temperature-dependentspecific heat cv(T ) by
ε(T ) =
∫ T
TR
cv(T )dT (2.5)
with a reference temperature TR . The radiation r followsfrom the equations of radiative transfer. In simpli-fied models, the radiation is modeled by the Stefan–Boltzmann law and depends only on temperature r(T ).
In MHD, the Lorentz force has typically been trans-formed into the divergence of the Maxwell tensor andOhmic heating is eliminated by considering the totalenergy balance. After this reformulation, the equationsfor density, velocity and magnetic field read
∂tρ+ div ρv = 0,
∂tρv + div
(ρvv +
(p+
1
2μ0B2
)I − 1
μ0BBT
)
= div(2μ(T ) grad v),
∂tB + div(B vT − vBT
)= − curl
(1
σ(T )curlB
), (2.6)
and for the total energy
Etot = ρ
(ε(T ) +
1
2v2
)+
1
2μ0B2, (2.7)
Modeling of electric arcs 3
including internal, kinetic and magnetic energy, we havethe total energy balance
∂tEtot + div
((Etot + p+
1
2μ0B2
)v − 1
μ0B(B · v)
)
= div( 1
μ0σ(T )B × curlB + 2μ(T )v · grad v
+λ(T ) gradT)
+ r(T ), (2.8)
which serves as an equation for temperature.The equations above represent a strongly coupled
system to describe the plasma flow in an electric arc.Usually, the mechanical equations and the heat transferare split from the electrodynamic part in a weak couplingapproach, for example in Zheng et al. (2004). In thefollowing, we investigate a different coupling in themagnetic diffusion, and heat transfer is strongly coupled.
2.2. Non-convective case
When neglecting the flow of the plasma v ≡ 0, theremaining relevant equations are
∂tB = − curl
(1
μ0σ(T )curlB
), (2.9)
∂t
(ρε+
1
2μ0B2
)= div
( 1
σ(T )B × curlB
+ λ(T ) gradT)
+ r(T ), (2.10)
for the magnetic field and temperature. These equa-tions model the dissipation of the plasma, while theremainder of (2.6) and (2.8) represent the convection ofan ideal non-dissipative plasma. The second equationcan be reduced to a diffusion equation for temperaturewhen eliminating the magnetic energy. We find the non-convective model for plasma
∂tB = − curl
(1
σ(T )curlB
),
ρcv(T )∂tT = div (λ(T ) gradT ) +1
σ(T )μ0(curlB)2 + r(T )
(2.11)consisting of two nonlinear diffusion-reaction equations.These equations describe the interplay of nonlinear mag-netic and temperature diffusion coupled by the electricalconductivity σ(T ) and linked to Ohmic heating andradiation.
2.3. Model discussion
2.3.1. Electric conductivity. The essence of the modeldescribed here lies in the choice of the temperature-dependent electric conductivity σ(T ). This functionmodels the neutral gas, the ionization and full plasmabehavior over the whole temperature range. While theconductivity is well defined for temperatures which pro-duce a sufficient degree of ionization, it is less simple forlow temperature where the medium is essentially neutral.
Model
Real data
Figure 1. (Colour online) Comparison of electric conductivitygiven by the model (2.12)/(2.13) and realistic data fromHuguenot (2008) with σ0 = 104 A (Vm)−1 and T0 = 3000 K.
When modeling electric arcs, the conductivity typicallydepends not only on temperature but also on the electricfield present in the gas (see Raizer 1991). An electricfield induces some ionization even at low temperatureand produces a non-vanishing electric conductivity.
In this paper, we simplify the situation and assume apure temperature dependence for σ, which is very smallbut non-vanishing for low temperature. We think of abackground electric field that justifies this assumption.As a model for conductivity, we will employ the erf-typefunction
s(T , σmin, σmax, T , ε) = σmin +σmax − σmin
1 − erf(T0−TεT0
)
×(
erf(T − T
εT0) − erf
(T0 − T
εT0
)),
(2.12)
which is controlled by a minimum σmin at temperatureT0, a maximum conductivity σmax for large temperatures,a transition temperature T , and a transition slope ε−1.Figure 1 compares the model with
σmin = 10−7σ0, σmax = σ0, T = 4T0, ε = 1 (2.13)
to realistic measurements for sulfur hexafluoride (SF6)gas obtained from Huguenot (2008). The electric con-ductivity spans at least seven orders of magnitude andthe qualitative behavior is reasonably captured by themodel. Below, we will use different values for σmin/σmax
as well as further reduced models for σ(T ) to obtainqualitative insight into the behavior of electric arcs.
2.3.2. Energy dissipation. In both full MHD and the non-convective model (2.11), energy is dissipatedthrough heat conduction and radiation. In high-pressurearcs, radiation is a major contribution to energy loss. Ittypically occurs in two variants, the local energy lossby a Stefan–Boltzmann law (optically thin medium), oran additional contribution to heat conduction (opticallythick medium). In order to reduce complexity and allowa transparent qualitative insight, we will neglect localenergy loss in this paper. To compensate the energydissipation, we will assume a larger heat conductivityinstead. In this way, we consider only a single energy
4 D. Wright et al.
dissipation mechanism. Furthermore, the heat conduct-ivity λ is assumed to be independent of temperature, thatis λ = const. In comparison with the strong dependenceof electric conductivity, the variations of λ are negligible(see Huguenot 2008). The inclusion of local losses andnonlinearity of λ is left for future work.
2.3.3. Specific heat/density. For simplicity, we will ad-ditionally assume cv = const and ρ = const in thenon-convective model (2.11).
2.3.4. Scaling. All quantities including space and timeare scaled according to ψ/ψ0 with some choice of scalingfactor ψ0. The choice of the electric conductivity scale isadditionally influenced by the choice of the model as in(2.12). To reduce complexity, we assume
t0
μ0σ0x20
= 1,μ0σ0λ
ρ0cv= 1,
B20
μ0ρ0cvT0= 1,
μ0j0x0
B0= 1,
(2.14)
such that space and time scale follow the magnetic dif-fusion, magnetic and thermal diffusion scale identically,and the magnetic field occurs on the scale of thermalenergy. The current density scale is linked to the scaleof the magnetic field by Ampere’s law. Some examplevalues satisfying this scaling are given by
⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩
σ0 = 104 A
Vm, λ = 8 × 102 W
mK, ρ0 = 10−2 kg
m3,
cv = 103 J
kgK, T0 = 3 × 103 K, B0 = 0.2 T,
j0 = 3 × 106 A
m2, x0 = 5 × 10−2 m, t0 = 2.6 × 10−5 s,
(2.15)
which are realistic values for strong electric arcs in SF6
corresponding to Fig. 1 and (2.12)/(2.13). Only the heatconductivity is one to two orders of magnitude too largein order to compensate for the neglected radiation.
With (2.14), no characteristic dimensionless para-meters appear in (2.11). The behavior of the systemis influenced by the choice of the conductivity modelσ(T ) and the inhomogeneous boundary conditions.
2.3.5. Numerical issues. After investigating the non-convective model in this paper, we will solve the fullMHD equation in a future paper. The MHD systemconsists of a flow part and dissipation, where the flow isgoverned by the divergence expressions on the left-handside in (2.6)/(2.8) and the dissipation by the right-handside. Furthermore, the dissipation part is nothing butthe non-convective model in (2.11).
Based on the findings of this paper, we assume thatthe essential behavior of the arc is controlled by thedissipation, while the flow is mostly a reaction. Hence,special care and understanding of the dissipation will be
Electric arc
ext
Figure 2. (Colour online) Settings for an infinite axisymmetricarc column in a cylinder. The arc can be driven by an externalfield or a prescribed total current.
worthwhile when designing the numerical method forthe full equations in a subsequent paper.
3. Infinite arc column modelTo study the properties of the non-convective model(2.11), we first consider an infinite arc column of plasmawithin a walled cylinder of radius R. The setting isdisplayed in Fig. 2. We assume axisymmetry and as suchall field variables depend only on time and the radius,r ∈ [0, R]. The electric current density points solelyalong in the direction of the z-axis of the cylinder j =(0, 0, j(z))T . Hence, the only non-vanishing magnetic fieldcomponent is in the angular direction B(ϕ). The arc canbe driven either by a prescribed total current Iext or anexternal electric field Eext.
In the following, we want to study the basic math-ematical behavior of arc creation and extinction as de-scribed by the non-convective model above. As alreadydiscussed, the assumptions for the electric conductivitylimit the physicality of the model. Still, the model is richenough to provide interesting mathematical and physicalinsight.
In axisymmetry, with spatial dependence only on r,(2.11) read
∂tB(ϕ) = ∂r
(1
μ0σ(T )
1
r∂r(r B
(ϕ))
), (3.1a)
∂tT =λ
ρcv
1
r∂r (r ∂rT ) +
1
ρcvσ(T )(j(z))2, (3.1b)
for the angular magnetic field B(ϕ) (r, t) and temperatureT (r, t). The current density j(z) (r, t) is given by
j(z) =1
μ0 r∂r
(r B(ϕ)
)(3.1c)
and enters the equation for temperature as quadraticOhmic heating. The equations are homogeneous; hence,any driving force must come from boundary conditionsor initial conditions.
At the z-axis, r = 0, we assume even and odd sym-metry for temperature and angular magnetic field, that
Modeling of electric arcs 5
is
r = 0 : ∂rT (0, t) = 0, B(ϕ) (0, t) = 0, (3.2)
and we fix the temperature at the cylinder wall
r = R : T (R, 0) = T0 (3.3)
with some given temperature T0 also used as temperat-ure scale.
In the current-driven process, the total current has tosatisfy
Iext = 2π
∫ R
0
j(z)(r)r dr =2π
μ0
∫ R
0
∂r(r B(ϕ)) dr
=2πR
μ0B(ϕ)(R) (3.4)
from Ampere’s law. This gives a Dirichlet boundarycondition
r = R : B(ϕ) (R, 0) = μ0Iext
2πR, (3.5)
for the magnetic field. When modeling arc creation westart with homogeneous initial conditions
t = 0 : T (r, 0) = T0, B(ϕ)(r, 0) = 0; (3.6)
hence, the external current in the boundary conditionsrepresents the driving force of the process. This settingturns out to be numerically difficult. It also does notcorrespond to the physical process. Starting with homo-geneous initial conditions, an arc is created by applyingan external potential difference, by means of an electricfield.
In the potential-driven case, we assume an electricpotential satisfying Δϕ(ext) = 0 within the cylinder witha potential difference at infinity such that the electricfield is given by Eext = ∂zϕ
(ext). Symmetry and geometryimply that Eext is constant both along the axis of thecylinder and in radial direction. It may, however, dependon time. The electric field enters the Neumann boundarycondition for the magnetic field, which reads
r = R : ∂r(r B(ϕ))
∣∣r=R
= R μ0σ(T0)Eext. (3.7)
In the case of homogeneous initial conditions, the driv-ing force of the process is represented by the electricfield.
The Neumann boundary conditions require σ(T0) �=0, otherwise the system becomes homogeneous andthe initial conditions would remain unchanged. If theequations based on the vector potential for the magneticfield were used, the external electric field would enterthe equation as a source term and σ(T0) = 0 could beimposed. Additionally, it can be shown for the steadycase that the solution of (3.1) with (3.7) converges tothe vector potential solution for σ(T0) → 0. Hence, itwill be part of our model to assume a small but finiteelectric conductivity at the wall. The reason for stickingto the description based on the magnetic field stemsfrom the usage of the MHD equations (2.6)/(2.8) in thefull convective case.
4. Steady arc column solutionsAfter creating an arc with given electric field, a steadytemperature field across the channel will emerge thatbalances the Ohmic heating due to the current with heatdiffusion. The temperature solution in the steady situ-ation satisfies the nonlinear reaction–diffusion equation
1
r∂r (r λ ∂rT ) = −σ(T (r))E2
ext (4.1)
and the fields of current density and magnetic field canbe computed by
j(z)(r) = σ(T (r))Eext, B(ϕ)(r) = Eextμ0
r
∫ r
0
σ(T (r))r dr
(4.2)
from the temperature. Boundary conditions for temper-ature are given by
T (R) = T0, ∂rT |r=0 = 0 (4.3)
in accordance with the description above. In the fol-lowing, we will investigate (4.1) for different electricconductivity functions σ(T ) and electric fields Eext. Therelevant dimensionless parameter is
E :=
√E2
extR2σ0
λT0, (4.4)
which describes the ratio between Ohmic heating andheat conduction.
4.1. Linear conductivity
The simplest model for electric conductivity is assuminga linear dependence of the form
σ(T ) = σ0 + α(T − T0), (4.5)
with σ(T0) = σ0 > 0 and a slope α > 0. The reaction–diffusion equation for temperature (4.1) reduces to aBessel differential equation with exact solution
T (r) = T0
(1 +
E2
A
(J0(
√A r/R)
J0(√A)
− 1
)), (4.6)
where J0 is the zeroth Bessel function of the first kind.The conductivity slope α enters the solution togetherwith the electric field Eext, the heat conductivity λ and theradius of the cylinder R in the dimensionless parameter
A =αE2
extR2
λ. (4.7)
If the slope α goes to zero, the solution reduces to theparabola
T (r) = T0
(1 + E2 1
4
(1 −
( rR
)2)), (4.8)
with maximal temperature Tmax = T0(1 + E2/4) inthe center. For α > 0, that is A > 0, this maximaltemperature increases and becomes infinite when
√A
reaches the first zero of the Bessel function J0. Forlarger values A > 5.78..., a positive temperature field
6 D. Wright et al.
does not exist any longer, due to oscillations of theBessel function.
If we choose σ0 very small, this mathematical blowupbehavior represents the ignition of an arc. For smallslopes, the solution can be interpreted as a temperat-ure field induced by creepage current in a capacitorrepresented by the infinite cylinder in our case. If theconductivity increase relative to heat conduction, electricfield and domain size becomes too large, the nonlinearincrease in Ohmic heating cannot be compensated byheat conduction and the temperature tends to infinity.Note that this blowup is independent of the value ofσ0 > 0. The blowup occurs faster with higher conduct-ivity slope, electric field, larger domain size, and smallerheat conductivity.
4.2. Exponential conductivity
The linear model for electrical conductivity is extremelysimplified. If we consider the exponential function
σ(T ) = σ0 exp
(α
σ0(T − T0)
), (4.9)
we expect a more physical result. Here, the parameter αagain takes the role of a slope or increase factor of theexponential σ′(T0) = α. For α → 0, we have σ(T ) = σ0.The equation for temperature (4.1) can again be solvedanalytically and gives two different solutions
T1,2(r) = T0
×
⎛⎜⎝1 +
E2
Aln
⎛⎜⎝ 8
(4 − A∓ 2
√4 − 2A
)(A+
(4 − A∓ 2
√4 − 2A
)( rR)2
)2
⎞⎟⎠
⎞⎟⎠,
(4.10)
labeled T1,2 depending on whether the minus or plussign is used in the logarithm. As described above, thesolution is essentially influenced by the parameter A asgiven in (4.7). For the slope α → 0 (A → 0), the solutionT1 converges to the parabola for constant conductivitywhile T2 becomes infinite. For increasing slopes T1 growsand T2 shrinks and both solutions meet for A = 2. Incase of values A > 2, the temperature solution ceases toexist due to complex values in the logarithm.
If we consider the small branch T1, the interpretationis analogous to the linear model. For small slopes, thetemperature corresponds to a creepage current, andbeyond a slope or beyond a certain electric field given byA = 2 the heat conduction cannot balance the nonlinearOhmic heating. Note that in this case, the temperaturedoes not blow up. The maximal temperature reached bycreepage in this model is T1(0)|A=2 = T0(1+ln(16)E2/4).
The second branch T2 is more difficult to understandat this point.
For a given electric field, the exponential model yieldstwo steady solutions that correspond to a low temperat-ure capacitor solution and an arcing solution exhibitinghigher temperature and higher electric conductivity and
current. Which of these steady solutions is realizeddepends on the process.
4.3. Step-shaped conductivity
Both the linear and exponential models for electricconductivity predict a non-existence of solutions forstrongly increasing conductivity functions. This non-existence is unphysical and due to the assumptionsthat the conductivity may tend to infinity. If we usethe function σ(T ) = s(T , σmin, σmax, T , ε) from (2.12), wehave a step-shaped function that starts at σmin at T0
and increases with temperature rapidly over five ordersof magnitude before leveling off to a large constantconductivity σmax. We use
σmin = 10−3σ0, σmax = 102σ0, T = 4T0, and ε = 1
(4.11)
for the arc column model. This model assumes a some-what smaller range of values for electric conductivitythan real data show, but allows qualitative insight intoarcing behavior.
An analytical solution cannot be found. We study thismodel for various cases of the dimensionless parameterE from (4.4) by solving (4.1) with a shooting method.That is, we prescribe a zero gradient and an estimatedvalue T� for the temperature at r = 0 and integratethe ordinary differential equation (4.1) as initial valueproblem up to r = R. The estimated value T� has to bechosen such that T (R) = T0.
For E = 1, we find three different solutions as shownin Fig. 3. One solution shows very little temperatureincrease and conductivity as well as current stays verylow. As before, this solution (capacitor solution) isinterpreted as creepage current in a capacitor. In anothersolution, the channel is almost completely filled withhighly conducting plasma and correspondingly largetemperature (labeled in the remainder as burning solu-tion). The intermediate solution shows a conductiveregion only in the center of the channel and mediumtemperatures (the arcing solution).
When increasing the electric field, that is the para-meter E, the capacitor solution changes very little, whilethe temperature of the burning solution increases andof the arcing solution decreases. For a critical Ecrit =17.4, the arcing and capacitor solutions meet and beyondthat critical value only the burning solution continues toexist. This behavior is also described by the exponentialconductivity above, and the existence of the burningsolution is the result of a finite maximal conductivity inthe present case.
When decreasing the parameter E, the burning solu-tion shrinks and the arcing solution increases until bothsolutions meet and vanish such that for very low valuesof E only the capacitor solution exists.
4.4. Current–voltage characteristics
The behavior of the arc column becomes more clearwhen considering the diagram of the current–voltage
Modeling of electric arcs 7
Temperature Electric conductivity
Arcing
Arcing
BurningBurning
CapacitorCapacitor
Figure 3. (Colour online) Three possible stationary solutions of the non-convective arc model, labeled by ‘capacitor’ solution,‘arcing’ solution and ‘burning’ solution.
characteristic. For all solutions found for a given electricfield Eext, we can easily calculate the total current flowingthrough the arc column by
Itot = 2π
∫ R
0
j(z)(r)r dr = 2π Eext
∫ R
0
σ(T (r; Eext)) r dr,
(4.12)
where we indicated that the solution T depends on Eext
or, to be precise, on the parameter E. The characteristicEext = ψ(Itot) gives a relation between the current andthe electric field or equivalently the voltage for a givenlength of the column. We write the characteristic indimensionless parameters as
√E2
extR2σ0
λT0= ψ
(Itot
R√σ0λT0
). (4.13)
The function ψ, as obtained for the infinite arc column(4.1) with (4.11), is shown in Fig. 4 and exhibits a non-monotone behavior that is typical for measurements ofelectric arcs. For an intermediate value of the electricfield, the characteristic shows three different values forthe total current, which correspond to the capacitor,arcing, and burning solutions. In general, the stronglyincreasing branch of the characteristic for very lowcurrents represents the capacitor solution. The arcingsolution is found along the decreasing branch until aminimum electric field. Finally, the slowly increasingbranch gives the burning solution. The slope of the capa-citor and burning branch is almost constant, indicatingan Ohm relation with a resistivity coefficient determinedfrom the minimal and maximal electric conductivity inthe model. The critical value Ecrit = 17.4 mentionedabove can also be read off the figure.
The question remains what solution will be establishedin a time-dependent simulation. We will see that thedecreasing branch represents unstable processes.
Figure 4. (Colour online) Current–voltage characteristics for asteady infinite arc column in a cylinder parameterized withdimensionless electric field and total current. The typicalnon-monotone behavior is visible.
5. Unsteady arc columnWe now consider the time-dependent equations (3.1) forthe infinite arc column model
∂tB(ϕ) = ∂r
(1
μ0σ(T )
1
r∂r
(r B(ϕ)
)),
∂tT =λ
ρcv
1
r∂r (r ∂rT ) +
1
ρcvσ(T )(j(z))2,
where the current density is given by
j(z) =1
μ0 r∂r
(r B(ϕ)
)on a domain r ∈ [0, R] using the step-shaped electricalconductivity (2.12) with parameter values (4.11). Ini-tially, the magnetic field vanishes and the temperature isgiven by the boundary value T0 throughout the domain.No current is flowing. At t = 0, we assume an externalelectric field greater than zero, Eext, which enters themodel through the Neumann boundary condition (3.7).After observing several possible solutions in the steadycase above, it is interesting to study what solution (capa-citor, arcing, burning) will appear in an unsteady processand how they are connected. This will be discussed afterintroducing the numerical method.
8 D. Wright et al.
1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.00 0
2 2
4 4
6 6
Contours of current (left) and temperature (right)
Current (center)
Temperature (center)
Temperature (lateral)
Current (lateral)
-2
2
-1
3
0
0
1
1
1
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
Temperature T/T0
T/T0
Radius r/RTime t/(μ0σ0R
2)
Tim
e/t( μ
0σ
0R
2)
Tim
e/t( μ
0σ0R
2)
Current j/j0
j/j0
0 0.8 1.6 2.4 3.2 4.8 5.6 6.4 7.2
10-3
10-1
101
103
100
101
102
103
4
Figure 5. (Colour online) Time evolution of temperature and current in an infinite arc column driven by a fixed external electricfield. Right: contours of current and temperature over time and space. Left: time evolution of lateral and center values.
5.1. Numerical method
We discretize the domain r = [0, R] using grid sizeΔr and the spatial derivatives in (3.1) scaled accordingto (2.14) using second-order finite differences resultingin the semi-discretization
∂tB(ϕ)
∣∣i
≈ 1
Δr2
(1
σi+ 12
1
ri+ 12
(ri+1B
(ϕ)i+1 − riB
(ϕ)i
)
− 1
σi− 12
1
ri− 12
(riB
(ϕ)i − ri−1B
(ϕ)i−1
)), (5.1a)
∂tT |i ≈ 1
Δr2
(ri+ 1
2
ri(Ti+1 − Ti) −
ri− 12
ri(Ti − Ti−1)
)
+1
σi
(j(z)i
)2
, (5.1b)
where
j(z)i ≈ 1
2Δr
(ri+1
riB
(ϕ)i+1 − ri−1
riB
(ϕ)i−1
). (5.1c)
For small electrical conductivity, σ(T ), the equationsare very stiff and explicit methods are not competitive.We used TR–BDF2, a second-order L-stable implicitRunge–Kutta scheme, whose first stage is a trapezoidalstep and second stage is a second-order backward dif-ference step (see Bank et al. 1985).
5.2. Arc creation
The behavior of this process depends on the value of theexternal field Eext, that is the value of the parameter Ein (4.4). For small values, the model will solely establisha small creepage current and come to a steady stateaccording to the capacitor solution. This represents thefirst steep branch in the characteristic of Fig. 4.
If we start from homogeneous conditions with con-stant temperature and no current and choose an externalfield beyond the critical value Ecrit = 17.4, a capa-citor solution becomes inadmissible according to thearc characteristic. The time evolution with t ∈ [0, 7.5]of temperature and current is displayed in Fig. 5 for thecase
E = 20, (5.2)
which is slightly above the critical value. In the figure,temperature is scaled by the boundary value T0 andcurrent density by the scale
j0 =B0
μ0R=
1
R
√ρ0cvT0
μ0(5.3)
based on the magnetic field scale. On the left-hand side,the figure shows a contour plot of both current density(left) and temperature (right) in a space–time diagram.
Both levels of current density and temperature areshown in logarithmic scale. Several sharp edges arevisible in the contours both of temperature and currentindicating transition periods, for example during ‘arcignition’ and current diffusion. For a better understand-ing of the contours, time evolutions of the fields atparticular values of r, namely r = 0.95R (lateral) andr = 0 (center), are plotted on the right-hand side ofFig. 5.
Up to t ≈ 0.8, the system is heated from relativelylow conductivity and small currents. The blowup andfull formation of the arc happen in two stages betweent ≈ 0.8 and t ≈ 1.6. First, the current density in thecenter grows quickly over several orders of magnitude,heating the temperature in the center to an intermediatevalue. The high current density together with large tem-perature values then moves toward the boundary. Thehigh temperature level then leads to a highly conductingplasma, exhibiting a skin effect and concentrating the
Modeling of electric arcs 9
Contours of current (left) and temperature (right)
1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.00 0
2 2
4
6
8 8
10 10
12 12
14 14
4
6
Current (center)
Temperature (center)
Temperature (lateral)
Current (lateral)
1.0 8.07.06.05.04.03.02.0
-2 -1 0 11 1 1 10 0 0 0
Temperature T/T0
T/T0
Radius r/R Time t/(μ0σ0R2)
Tim
e/t(μ
0σ
0R
2)
Tim
e/t(μ
0σ
0R
2)
Current j/j0
j/j0
0 2 4 8 12
10-5
10-3
10-1
101
1
3
5
7
6 10 14
Figure 6. (Colour online) Time evolution of temperature and current in an infinite arc column driven by a fixed external electricfield, which is switched down to a small value at t = 30. Right: contours of current and temperature over time and space. Left:time evolution of lateral and center values.
current near the boundary. For a short time, this leadsto a higher lateral temperature value compared withthe center. However, the overall heating presumes andthe center temperature takes over again quickly. Fort > 1.6, the current density diffuses slowly toward thecenter while heating the plasma. The final steady state,which is not yet reached in the figure, corresponds to aburning solution in which essentially the whole channelis filled with highly conducting plasma.
Note that both current density and temperature in-crease to extremely high values which are unphysical.In reality, the arc is created by applying a voltage, thatis an external electric field, but once a significant totalcurrent is flowing the generator typically is not able tomaintain the voltage and reduces it accordingly.
5.3. Switch-down
We now consider the same process as in the previoussubsection, but switch down the external electric fieldshortly after the creation of the arc at t = 1.2. Toavoid numerical instabilities, we reduce the parameterE in the boundary conditions smoothly from E = 20to 0.05 in the time interval t ∈ [1.2, 1.5]. The valueE = 0.05 only admits a capacitor solution according tothe characteristic in Fig. 4.
The time evolution of this switching process is shownin Fig. 6. The figure uses the same layout as Fig. 5 withcontour plots in the space–time diagram on the leftand time evolutions of particular values on the right.Only the current density is shown logarithmically, whiletemperature uses natural scaling. The beginning of theprocess is identical to Fig. 5 except that the scaling ofthe time axis is reduced in the present case. The time ofthe switching is marked in the figure by arrows on thetime axes.
The arc does not vanish immediately after switch-ing down the electric field. The lateral current densitydecreases dramatically; however, the center value re-mains the same or is even increasing later. This leadsto a focusing of the arc in the center of the channelas also visible in the contour plot. A very weak skineffect is visible inside the plasma column. Similar tothe current density, the temperature strongly decreasestoward the boundary and concentrated in the center.The arc is burning with considerable strength for arelatively long time period and steady temperature pro-file. It maintains itself through sufficient energy storedin the temperature profile, even though the only steadysolution for the acting electric field would be a capacitorsolution.
With time, the temperature decreases very slowly. Ata certain time, the temperature drops below a valuethat is needed to maintain the arc. This depends on thefunctional relation for the electric conductivity. Afterpassing this point, the current density almost instantlyfalls to the value given by creepage current correspond-ing to the very low external electric field. Similarly, thetemperature dissipates to a very small value slightlyabove the boundary condition.
5.4. Stability
In the above scenario, the capacitor was the only possiblesolution, which had to be assumed for long-time station-ary state. In this section, we consider the same settingagain and create the arc from homogeneous initial valueswith an electric field E = 20, as above, but switch downto the value E = 0.75, which admits all three solutions(capacitor, arcing, burning) (see Fig. 4). We will considerthree different switching times t = 1.1, 1.2, 1.3, which allproduce different solutions.
10 D. Wright et al.
Projected
SolutionArcing
Capacitor
Figure 7. (Colour online) Arc evolutions in the projectedtemperature solution space. The arc is driven by a strongexternal field, which is switched down to an intermediate valueafter some time. Different times result in different solutionpaths.
To allow a better comparison, we display the solutionpaths of the temperature field. For this, the temperaturesolution T (r, t) at a certain time t is projected onto thepoint
τ(t) := (T (r�, t), T (0, t)). (5.4)
This reduces the infinite dimensional temperature fieldto a 2D point given by the center value at r = 0 and alateral value at r� = 0.6R. The curve τ(t) represents theprojected solution trajectory and can easily be plottedfor different processes. Note that the stationary solutionsare represented by points τ(capacitor), τ(arcing), and τ(burning)
in the projected temperature solution space.Figure 7 shows the projected solution space and four
different temperature evolutions. All solutions start withhomogeneous temperature at the point τ(0) = (1, 1). Onetrajectory is given by the unswitched process describedin Sec. 5.2, shown as red curve (mostly hidden by othercurves) leaving the plot in Fig. 7 toward the right. Theinitial heating phase and ignition is visible when thecenter temperature increases rapidly. Afterward, bothtemperatures steadily increase toward the only possiblestationary solution (burning) that is situated to the rightfar outside the plot.
The other three trajectories in Fig. 7 show processesthat are started identically to the unswitched case, butswitched down at different times. Hence, the solutionsfollow the red curve initially until the switching time.After switching to the lower electric field E = 0.75, thetopology of the solution space changes and three sta-tionary points arise representing the capacitor solutionnear (1, 1), the arcing solution in the middle of the plot,and a burning solution that lays outside of the plotbeyond the upper right corner. Which solution is finallyapproached depends on the switch-down time, that is,how much energy has been inserted into the systemdue to the large electric field. In the case of the twolater times t = 1.2 and 1.3, the solution approaches theburning solution. On the other hand, when switching
x
−3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
y -20
2
z
−2−1
01
2
Figure 8. (Colour online) Sketch of geometry for a 3Daxisymmetric simulation.
0 0.5 1 1.5 2 2.5 3 3.5 4
× 104
0
2000
4000
6000
8000
10000
12000
14000
16000
T (K)
σ(T
)(S
/m
)
Figure 9. (Colour online) The electrical conductivity σ(T ) usedin the 3D axisymmetric simulations, according to Huguenot(2008).
down earlier at t = 1.1, the final solution will be thecapacitor solution, that is the arc extinguishes due tolack of energy. The arcing solution is never assumedas final solution as it turns out to be unstable. This isknown in electric engineering from considerations of thecurrent–voltage characteristic (see Maecker 1951; Rieder1967). Here, it becomes visible from the mathematicalbehavior of a coupled system of nonlinear reaction–diffusion equations.
Note, however, that the arcing solution is approachedfrom all three trajectories before they bend away. Thestationary arcing point is a saddle point in the solu-tion space and consists of an attractive and repulsivesubmanifold shown as dashed lines in Fig. 7. The neigh-borhood of the saddle point is relatively flat; hence,the solution trajectories cross this region very slowly.For the lowest trajectory, the situation is similar to theswitch-down case discussed above in Sec. 5.3.
After switching down, the arc exists for a fairly longtime in which it may be observed as stable. However,
Modeling of electric arcs 11
Figure 10. (Colour online) The electric potential ϕ and applied external electric field ‖Eext‖, the gray rectangles in the right-handfigure depict the position of the contacts.
this is merely a quasi-stability due to the flatness of theneighborhood of the saddle point forcing the solutionto change slowly. After sufficient time, all trajectoriesleave the vicinity of the arcing point either to producea strong arc or to face extinction. The case decision ismade by the amount of energy present in the system.
6. 3D axisymmetric simulationsUsing the insights gained by studying the infinite arccolumn, we now solve the model numerically in 3Daxisymmetric cylindrical coordinates mimicking contactgeometry typically found in high-voltage circuit breakersand the real material coefficients of SF6 gas as givenin Huguenot (2008).
In 3D axisymmetric cylindrical coordinates, the fieldshave no angular dependence, that is B(ϕ) ≡ B(ϕ)(t, r, z)and T ≡ T (t, r, z), and the equations for the scalednon-convective model (2.11) become
∂tB(ϕ) = ∂r
(1
σ(T )
1
r∂r
(rB(ϕ)
))− ∂z
(1
σ(T )∂zB
(ϕ)
),
(6.1a)
∂tT = κ1
r∂r (r∂rT ) + κ∂z (∂zT ) +
1
σ(T )|j|2 , (6.1b)
where j = ( −∂zB, 0, r∂r(rB) )T. Note that despite thefact we are considering the model in two dimensions, ifwe assume B(r) = 0 and B(z) = 0 at t = 0, it suffices toonly consider the evolution of the ϕ-component of themagnetic field and the equation for B becomes scalar.
6.1. Contact modeling and arc initialization
We consider (6.1) in a rectangular domain (r, z) ∈[0, R] × [−Z,Z] and model the initialization of the arcand the position of the contacts as shown in Fig. 8 withan appropriate choice of boundary conditions. Initially,we assume the process is potential driven and specifyinhomogeneous Neumann boundary conditions for B(ϕ)
in terms of the external electric field Eext = (E(r)ext, 0, E
(z)ext)
T
as
∂zB(ϕ)|z=±Z = μ0σ(T0)E
(r)ext, (6.2)
∂r(rB(ϕ))|r=R = Rμ0σ(T0)E
(z)ext. (6.3)
Note that the ϕ-component of the external electricfield is zero since the electric potential has no angulardependence in axisymmetry (see (6.5)). When the totalcurrent in the arc, as given by (3.4),
I(t, z) = 2π
∫ R
0
rj(z)(r) dr = 2π
∫ R
0
r∂rB(ϕ)(t, r, z)
dr= 2πRB(ϕ)(t, R, z)
is greater than some specified value, that is I > Imax, wechange the boundary conditions to current driven, thatis we specify Dirichlet boundary conditions for B(ϕ) onthe boundary in terms of the attained current I
B|z=±Z =I
rat wall,
∂zB|z=±Z = 0 at contacts,
∂r(rB)|r=R =I
Rat outside wall.
We denote the time at which the switch from potentialto current driven occurs as ts. The switch from potentialto current driven allows us to initialize simulationseffectively. This was previously a difficult problem whichrequired an extremely high initial temperature. Thearc was then considered to burn in the part of thedomain where the gas had not cooled (see Kumar2009). Although it is a step toward modeling electricarc formation in circuit breakers, it cannot and does notaim to fully capture the true physical process of ignition.The initial conditions used in numerical simulations aregiven in Sec. 6.3.
The boundary conditions for B(ϕ) together with suit-able mixed boundary conditions for the temperature
12 D. Wright et al.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
2000
4000
6000
8000
10000
12000
t (ms)
T(K
)
r = 0.10r = 0.15r = 0.20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.70
1000
2000
3000
4000
5000
6000
t (ms)
J(k
A/m
2)
r = 0.10r = 0.15r = 0.20
Figure 11. (Colour online) The initial magnetic field B and current density J . The initial temperature is constant T = 300 K.
given by
T (t, r, z) = T0 at all none-contact boundaries,
∂zT |z=±Z = 0 at contacts,
model the position of the contacts. Note the boundaryconditions for the temperature are independent of thepotential- or current-driven mode.
The values of T and B(ϕ) at r = 0 in both current- andpotential-driven modes are again specified by considerthe even, respectively, odd symmetry of the temperatureand magnetic fields along the z-axis of the cylinder.
In order to understand why (6.2) models the positionsof the contacts appropriately for the potential-drivencase, it remains to explain how we specify the compon-ents of the electric field Eext. To calculate the externalelectric field, we specify the potential across the contactsin terms of Δφ0, solve the Poisson problem
Δφ = 0, (6.4a)
φ = ±Δφ0 at the contacts, (6.4b)
∂φ = 0 elsewhere, (6.4c)
and calculate the electric field according to
Eext = − gradφ. (6.5)
6.2. Discretization
As for the infinite arc column, we use second-order finitedifferences to discretize the spatial derivatives. The gridis Cartesean with the i-index for the radial and the j-index for the z-coordinate. The semi-discretization ofthe axisymmetric non-convective model (6.1) becomes
∂tB(ϕ)
∣∣i,j
≈ 1
Δz2
(1
σi,j+ 12
B(ϕ)i,j+1 − 2
σi,jB
(ϕ)i,j +
1
σi,j− 12
B(ϕ)
i,j− 12
)
+1
Δr2
(1
σi+ 12 ,j
1
ri+ 12
(ri+1B
(ϕ)i+1,j − riB
(ϕ)i,j
)
− 1
σi− 12 ,j
1
ri− 12
(riB
(ϕ)i,j − ri−1B
(ϕ)i−1,j
))(6.6a)
for the magnetic field equation and
∂tT |i,j ≈ 1
Δz2
(κi,j+ 1
2Ti,j+1 − 2κi,jTi,j + κi,j− 1
2Ti,j−1
)
+1
Δr2
(ri+ 1
2
riκi+ 1
2
(Ti+1,j − Ti,j
)
−ri− 1
2
riκi− 1
2
(Ti,j − Ti−1,j
))
+1
σ(Ti)
((j(r)i,j
)2+
(j(z)i,j
)2), (6.7a)
where
j(r)i,j ≈ − 1
2dz
(B
(ϕ)i,j+1 − B
(ϕ)i,j−1
), (6.7b)
j(z)i,j ≈ 1
2Δr
(ri+1
riB
(ϕ)i+1,j − ri−1
riB
(ϕ)i−1,j
), (6.7c)
for the energy balance, where the current componentsare given by
j(r)i,j ≈ − 1
2dz
(B
(ϕ)i,j+1 − B
(ϕ)i,j−1
), (6.8a)
j(z)i,j ≈ 1
2Δr
(ri+1
riB
(ϕ)i+1,j − ri−1
riB
(ϕ)i−1,j
). (6.8b)
The resulting system of ordinary differential equationsis solved by the second-order implicit TR–BDF2 time-integration scheme (Bank et al. 1985).
6.3. Numerical experiment: hollow oblique contacts
We now consider the solution of (6.6)–(6.8a) usingκ(T ) ≡ const and electrical conductivity σ(T ) ∈ [0.2, 1.7×104] fitted to real data as in Fig. 9 (Huguenot 2008;Kumar 2009). The computational domain is [0, 0.5] ×[−0.1, 0.1] m, where the contacts are of width 0.05mat 0.15–0.20m at the bottom of the domain and 0.10–0.15m at the top of the domain; for a schematic of thedomain with contacts, see Fig. 10.
We initially specify a temperature of T0 = 300Kand apply an electric potential of 240 kV, that is φ0 =±120 kV at the contacts. The electric potential and elec-tric field as given by the solution to (6.4) and (6.5) with
Modeling of electric arcs 13
Figure 12. (Colour online) The evolution of T (t, r, z = 0) and ‖J(t, r, z = 0)‖ at the points (r, z) = (0.10, 0), (0.15, 0), (0.20, 0) insidethe circuit breaker. The values of r correspond to the inside of the top contact, the midpoint between the two contacts, and theoutside of the bottom contact, respectively. The vertical line at t = 0.323 ms represents the time of switch-down. The time ofswitching from potential to current driven, ts, occurs at t = 3.14 × 10−4 ms and cannot be seen in this plot.
Figure 13. (Colour online) The temperature T and current density J before switching from potential to current driven att = 1.57 × 10−4 ms (t = 0.5 ts).
Figure 14. (Colour online) The temperature T and current density J at the point of switching from potential to current drivenat t = 3.14 × 10−4 ms.
these values are shown in Fig. 10. The initial conditionfor B(ϕ) is calculated by solving the stationary prob-lem for (6.6) with the potential-driven boundary condi-tions (6.2), shown in Fig. 11. Due to the non-constant
value of B(ϕ), there is an initial creepage current betweenthe contacts (see Fig. 12). Furthermore, we specify thatthe simulation switches from potential driven to currentdriven when the total current in the circuit breaker has
14 D. Wright et al.
Figure 15. (Colour online) The temperature T and current density J at t = 3.14 × 10−3 ms (t = 10 ts).
Figure 16. (Colour online) The temperature T and current density J at t = 3.14 × 10−2 ms (t = 100 ts).
Figure 17. (Colour online) The temperature T and current density J immediately before shutdown at t = 0.322 ms.
reached Imax = 200 kA. In this example, the switch tocurrent driven occurs at ts = 3.14 × 10−7 s and afterthe solution has attained a steady state, we switch downthe electric arc by reverting back to potential-drivenboundary conditions, but with negligible Eext, whichoccurs at t = 3.22 × 10−4 s.
Clearly, the potential-driven phase occurs on a verydifferent scale to the rest of the solution, also de-
picted in Fig. 12. We see that in Figs. 13 and 14the current density is initially concentrated around thecorners of the contacts before forming a narrow arcof very high current density, which we consider tobe modeling the skin effect. We also see that beforeswitching to current driven, very high temperaturesare already attained within the arc where current isflowing.
Modeling of electric arcs 15
After switching to current driven, the total currentin the arc is held constant and the current densitywithin the arc diffuses inward toward the z -axis andis more evenly distributed across the contacts. We seefrom the temperature that the gas within the device heatsup accordingly, leading to higher electrical conductivity.This leads to development of a well-formed electric arcburning between the contacts, as can be seen in Figs. 14–16. Note the largest value attained by the magnetic fieldis B = 0.286 T at t = 0.322 ms, immediately before theshutdown of the electric arc (see Fig. 17).
For the shutdown, the simulation is switched to avoltage-driven situation at t = 0.322, but with a van-ishing external voltage. In contrast to the 1D case, nopseudo-stable arc is observed, instead the arc vanishesquickly after the switch (see Fig. 12). However, thetemperature remains high, which results in re-creationof the arc in an alternating current situation.
7. ConclusionWe presented a comprehensive study of the mathem-atical features of the magnetothermal system that de-scribes electric arcs. The equations, based on the stronglycoupled MHD system for plasma flows, consist of Ohmicheating and nonlinear electric conductivity, which arethe core components of arcing processes. The non-uniqueness and stability features of the model can belinked to actual physical properties of the electric arc.Even though the model is very rough and does notinclude important features like radiation or dielectricbreakdown, it allows a qualitative description of arccreation and extinction.
The paper also presents 3D axisymmetric simulationsof the model, which use strongly coupled numericalmethods. In contrast to weakly couple electrodynamicsto the heat transfer, the approach of this paper solvesheat transfer and magnetic diffusion in a single timeintegration. Future work will include the coupling of theactual flow and an external electric network. The aim isto contribute to electric arc simulations, for example inthe case of very high current circuit breakers.
ReferencesBank, R. E., Coughran, W. M., Fichtner, W., Grosse, E. H.,
Rose, D. J. and Smith, R. K. 1985 Transient simulation
of silicon devices and circuits. IEEE Trans. Computer-Aided
Des. 4, 436–451.
Cassie, A. M. 1939 Arc rupture and circuit severity. A
new theory. In: Conference des Grands Reseaux Electriques ,
Electrical Research Association Technical Report, E.R.A.
Vol. 108, Springer, 14 pp.
Delmont, P. and Torrilhon, M. 2012a Convective
magnetohydrodynamical modeling and simulation of
electric arcs. In: Europhysics Conference Abstracts Vol.
36F ISBN 2-914771-79-7. 39th EPS Conference on Plasma
Physica & 16th International Congress on Plasma Physics ,
2–6 July 2012, Stockholm, Sweden (ed. S. Ratynsakaya, L.
Blomberg, A. Fasoli) p. P1.138.
Delmont, P. and Torrilhon, M. 2012b Magnetohydrodynamical
modeling and simulations of electric arc extinction in a
network. In: CD-ROM Proceedings of the 6th European
Congress on Computational Methods in Applied Sciences and
Engineering (ECCOMAS 2012) (ed. H. J. Rammerstorfer, F.
G. Eberhardsteiner and J. Bhm). Vienna, Austria: Vienna
University of Technology.
Finkelnburger, W. and Maecker, H. 1956 Elektrische Bogen
und thermische plasmen. In: Handbuch der Physik (ed.
S. Flugge). Berlin: Springer, pp. 254–440.
Garzon, R. D. 2002 High Voltage Circuit Breakers: Theory and
Design . New York: CRC Press.
Gleizes, A., Gonzales, J. J. and Freton, P. 2005 Thermal plasma
modelling. J. Phys. D: Appl. Phys. 38, 153–183.
Goedbloed, J. P. H. and Poedts, S. 2004 Principles
of Magnetohydrodynamics With Applications to Laboratory
and Astrophysical Plasmas, Cambridge University Press,
Cambridge, UK.
Huguenot, P. 2008 Axisymmetric high current arc simulations
in generator circuit breakers based on real gas
magnetohydrodynamics models. PhD thesis, ETH.
Kosse, S., Wendt, M., Uhrlandt, D., Weltmann, K.-D. and
Franck, Ch. 2007 Simulation of moving arcs. Pulsed Power
Conference pp. 1013–1017.
Kumar, H. 2009 Three dimensional high current arc
simulations for circuit breakers using real gas resistive
magnetohydrodynamics. PhD thesis, ETH.
Maecker, H. 1951 Der elektrische lichtbogen. Ergebnisse der
Exacten Naturwissenschaft 25, 293–358.
Maximov, S., Venegas, V., Guardado, J. L. and Melgoza,
E. 2009 A method for obtaining the electric arc
model parameters for SF6 power circuit breakers. IEEE
EUROCOM 2009 International Conference Devoted to
the 150th Anniversary of Alexander Popov, St. Petersburg,
Russia, 13–23 May 2009, pp. 458–463. Red Hook, NY:
Curren Associates.
Mayr, O. 1943 On the theory of electric arc and its extinction.
Electrotechnische Zeitschrift ETZ 64, 645–652.
Mayr, O. 1955 On the stability limits of switching arcs. Applied
Scientific Research, Section B (Electrophysics, Acoustics,
Optics, Mathematical Methods) 5, 241–247.
Raizer, Yu. P. 1991 Gas Discharge Physics . New York:
Springer.
Rieder, RW. 1967 Plasma und Lichtbogen . Berlin: Vieweg.
Torrilhon, M. 2007 Uniqueness and stability of a reduced arc
model. Proc. Appl. Math. Mech. 7(1), 1141301–1141302.
Tseng, K.-J., Wang, Y.-M. and Vilathgamuwa, D. M. 1997
Experimentally verified hybrid cassie-mayr electric arc
model for power electronic simulations. IEEE Trans. Power.
Elect. 12(3), 429–437.
Van Der Sluis, L. 2001 Transients in Power Systems . New
York: John Wiley & Sons.
Zehnder, L., Kiefer, J., Braun, D. and Schonemann, T.
2002 SF6 generator circuit-breaker system for short-circuit
currents up to 200 kA. ABB Technology Review 3, 34–40.
Zheng, S., Liu, J.-Y., Gong, Y. and Wang, X.-G. 2004 Two-
dimensional numerical simulation on evolution of arc
plasma. Vacuum 73, 373–379.